Abstract. In
this talk, we will study complete families of smooth space curves, that
is complete subvarieties of the Hilbert scheme of smooth curves in P3.
On the one hand, we will construct non-trivial examples of such
families. On the other hand, we obtain necessary conditions for a
complete family of smooth polarized curves to induce a complete family
of non-degenerate smooth space curves. Both results rely on the study of
the strong semistability of certain vector bundles.